Solution of Different Types of Economic Load Dispatch Problems Using a Pattern Search Method

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Solution of Different Types of Economic Load Dispatch

Problems Using a Pattern Search Method

J. S. Al-Sumaita; J. K. Sykulskia; A. K. Al-Othmanb

a_{Electrical Power Engineering Group, University of Southampton, Highfield,}

Southampton, United Kingdom

b_{Electrical Engineering Department, College of Technological Studies, Alrawda,}

Kuwait

Online Publication Date: 01 March 2008

To cite this Article: Al-Sumait, J. S., Sykulski, J. K. and Al-Othman, A. K. (2008) 'Solution of Different Types of Economic Load Dispatch Problems Using a Pattern Search Method', Electric Power Components and Systems, 36:3, 250 - 265

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ISSN: 1532-5008 print/1532-5016 online DOI: 10.1080/15325000701603892

Solution of Different Types of Economic Load

Dispatch Problems Using a Pattern Search Method

J. S. AL-SUMAIT,

1

_{J. K. SYKULSKI,}

1

_and

A. K. AL-OTHMAN

2

1_{Electrical Power Engineering Group, University of Southampton,}

Highfield Southampton, United Kingdom

2_{Electrical Engineering Department, College of Technological Studies,}

Alrawda, Kuwait

Abstract Direct search (DS) methods are evolutionary algorithms used to solve constrained optimization problems. DS methods do not require information about the gradient of the objective function when searching for an optimum solution. One such method is apattern search (PS) algorithm. This study presents a new approach based on a constrained PS algorithm to solve various types of power system economic load dispatch (ELD) problems. These problems include economic dispatch with valve point (EDVP) effects, multi-area economic load dispatch (MAED), companied economic-environmental dispatch (CEED), and cubic cost function economic dispatch (QCFED). For illustrative purposes, the proposed PS technique has been applied to each of the above dispatch problems to validate its effectiveness. Furthermore, convergence characteristics and robustness of the proposed method has been assessed and inves-tigated through comparison with results reported in literature. The outcome is very encouraging and suggests that PS methods may be very efficient when solving power system ELD problems.

Keywords ELD, DS method, PS method, evolutionary algorithms, optimization

1. Introduction

Scarcity of energy resources, increasing power generation costs, and ever-growing de-mand for electric energy necessitates optimal economic dispatch (ED) in today’s power systems. The main objective of ED is to reduce the total power generation cost while satisfying various equality and inequality constraints. Traditionally, in ED problems, the cost function for generating units has been approximated as a quadratic function.

A variety of optimization techniques has been applied in solving ELD problems. Some of these techniques are based on classical optimization formulations, while others are based on artificial intelligence methods or heuristic algorithms. Many references present the application of classical optimization methods, such as linear or quadratic programming, to solve ELD problems [1, 2]. Classical optimization methods are reported to be highly sensitive to the selection of the starting points and sometimes converge to a

Received 12 October 2006; accepted 13 July 2007.

Address correspondence to Dr. Abdrulrahman Al-Othman, Electrical Engineering Department, College of Technological Studies, P.O. Box 33198, Alrawda, 73452, Kuwait. E-mail: ak.alothman@ paaet.edu.kw

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local optimum or diverge altogether. Linear programming methods are fast and reliable, but have disadvantages associated with the piecewise linear cost approximation. Non-linear programming methods are known to have problems of convergence and algorithmic complexity. Newton-based algorithms struggle with handling a large number of inequality constraints [3]. Methods based on artificial intelligence techniques, such as artificial neural networks, are also presented [4, 5]. Heuristic search techniques, such as particle swarm optimization [3] and genetic algorithms [6], have also been successfully applied to ELD problems. Finally, hybrid methods have been developed, where the conventional Lagrangian relaxation approach, a first-order gradient method, and multi-pass dynamic programming (DP) are combined in [7].

Recently, a particular family of global optimization methods, originally introduced and developed by researchers in the 1960s [8], has attracted special attention. Known as DS methods, they are simply designed to explore a set of points around the current position, looking for a location that has a better objective value than the existing one. This family includes PS algorithms, simplex methods (different from the simplex used in linear programming), Powell optimization, and others [9].

DS methods, in contrast to more standard optimization methods, are often called derivative-free as they do not require information about the gradient or higher derivatives of the objective function to search for an optimal solution. Thus, DS methods are a popular choice to solve non-continuous, non-differentiable, and multimodal (i.e., multiple local optima), optimization problems. Since the ED is one such problem, the proposed method appears to be a good candidate to achieve efficient solutions [10].

The main objective of this study is to introduce the use of a PS optimization technique in application to the power system ELD. The performance, effectiveness, and robustness of the proposed method are assessed via intensive testing and comparison with results reported in literature. The article is organized as follows: Section 2 introduces the problem formulation, Section 3 presents a description of the proposed PS algorithm, analysis and test results are presented in Section 4, followed by concluding remarks.

2. Problem Formulation

Generally, depending on the type of the ELD considered, the traditional formulation is basically a minimization of summation of the fuel costs of the individual dispatchable generators, subject to the real power balanced with the total load demand as well as with the limits on generators’ outputs. The mathematical form of each type of ED problem considered in this study may be described as in the following subsections.

2.1. Mathematical Form of EDVP

The inclusion of the valve-point effects is advantageous and makes the modeling of the fuel-cost function more realistic [11–13]. However, the valve-point effects, which appear as a sinusoidal term added to the fuel-cost functions, introduce ripples to the heat-rate curve, and, therefore, create more local minima in the search space.

The objective function of an EDVP is to minimize F , where F is given by

F D

N

X

i D1

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The incremental fuel-cost function of the generation units with valve-point loadings are represented as follows [11]: Fi.Pi/ D aiPg i2 C biPg iC ci C jeisin.fi .Pg i.min/ Pg i//j (2) subject to N X i D1 Pg i D PDC PL (3) Pg i.min/< Pg i < Pg i.max/; i 2 Ns; (4) where

F is the system overall cost function;

N is the number of generators in the system;

ai, bi, and ci are the constants of fuel function of generator number i ;

ei, fi are the constants of the valve-point effect of generator number i ;

Pg i is the active power generation of generator number i ;

PD is the total power system demand;

PLis the total system transmission losses;

Pg i.min/ is the minimum limit on active power generation of generator i ; Pg i.max/ is the maximum limit on active power generation of generator i ; and Ns is the set of generators in the system.

2.2. Mathematical Form of MAED

In a MAED problem, the objective is to determine the most economical generation level in a given area and the power interchange between the areas that minimize the overall operation cost while satisfying a set of constraints as follows [14]:

minhXF .Pg i/ C X G.tj k/ i (5) subject to X i 2˛m Pg iC X j 2ˇm tkj X j 2ˇm tj k DmD 0 (6) Pg i.min/ Pg i Pg i.max/ (7) tj k.min/ tj k tj k.max/; (8) where

F .Pg i/is the fuel-cost function of generator Pg i,

G.tj k/is the cost of transmission of line tj k,

tj k is the economic flow on the tie line from area j to k,

˛m is the set of generating units in area m,

ˇm is the set of tie lines in area m,

Pg i is the active power generation of generator number i ,

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PL is the total system transmission losses,

Pg i.min/ is the minimum limit on active power generation of generator i , Pg i.max/ is the maximum limit on active power generation of generator i , tj k.min/ is the minimum limit on active power generation of generator i , and tj k.max/ is the maximum limit on active power generation of generator i .

2.3. Mathematical Form of CEED

The primary objective of ED can be extended to take into consideration the environmental impact of power generation due to the emission of different pollutants that could inflict harm to the environment [15, 16]. The emission-cost function, just like the fuel-cost function in traditional ED, is modeled as a second-order polynomial

E D

N

X

i D1

diPg i2 C hiPg i C ui (kg/h) (9)

where di, hi, and ui are emission coefficients of generator i . Thus, the overall objective

function for the combined emission–economic dispatch is

min F C.1 w/ w E (10)

where w is a weighting factor, w 2 Œ0; : : : ; 1, which represents the relative importance or the trade-off between the fuel-cost function F and the emission-cost function E. The objective function of Eq. (10) is minimized, subject to the constraints given by Eqs. (3) and (4), where the transmission losses are given by

PLD N X nD1 N X kD1 Pg nBn;kPgk (11)

where B is a matrix of loss coefficients and Pg nand Pgk are the active power generation

of generators n and k, respectively.

2.4. Mathematical Form of QCFED

Traditionally, the cost function of the generating units is approximated as a quadratic function. A crucial issue in ED studies is to determine the order and approximate the coefficients of the polynomial used to model the fuel-cost function [17]. This issue is particularly important in terms of reducing the error between the approximated polyno-mial, along with its coefficients, and the actual operating cost. According to [17, 18], a third-order (cubic) polynomial is realistic to model the operating cost. For a generating unit with non-monotonically increasing cost curves, a cubic polynomial is usually used to obtain accurate dispatch results [19]. The third-order polynomial function is expressed as

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Downloaded By: [University of Southampton] At: 12:17 22 February 2008 subject to N X i D1 Pg i D PDC PL (13) Pg i.min/< Pg i < Pg i.max/; i 2 Ns; (14) where ˛i;1, ˛i;2, ˛i;3, and ˛i;4 are the coefficients of the cubic fuel-cost function.

System Losses Representation. It important to mention at this stage that the B-coefficients, or loss coefficients, have been adopted for the modeling of the system losses in the previous formulations. The representation of the system losses by B-coefficients is indeed constant and suitable for actual interpretation of the real power system losses under certain conditions. If the actual operating conditions are close to the base case where B-constants were computed, then the B-coefficients method should compute the system losses with reasonably high accuracy [20, 21]. In other words, the use of constant values for the loss coefficients in the transmission losses equation yields practically precise results when the coefficients are calculated for some average operating condition and if extremely wide shifts of load between plants or in total load do not occur. In practice, large systems are economically loaded by calculations based on just one set of loss coefficients that are sufficiently accurate throughout the daily variations of load on the system [21].

3. Overview of the PS Method

The PS optimization routine is an evolutionary technique that is suitable to solve a variety of optimization problems that lie outside the scope of the standard optimization methods. Generally, PS has the advantage of being very simple in concept, being easy to implement, and resulting in a computationally efficient algorithm. Unlike other heuristic algorithms, such as genetic algorithms (GA) [22, 23], PS possesses a flexible and well-balanced operator to enhance and adapt the global and fine tune the local search. A historic discussion of DS methods for unconstrained optimization is presented in [9], where a modern perspective on the classical family of derivative-free algorithms is given with focus on the development of DS methods.

The PS algorithm proceeds by computing a sequence of points that may or may not approach the optimal point. The algorithm starts by establishing a set of points called a mesharound the given point. This current point could be the initial starting point supplied by the user, or it could be computed from the previous step of the algorithm. The mesh is formed by adding the current point to a scalar multiple of a set of vectors called a pattern. If a point in the mesh is found to improve the objective function at the current point, the new point becomes the current point at the next iteration.

This may be better explained by the following:

First: The PS begins at the initial point X0 that is given as a starting point by the user.

At the first iteration, with a scalar D 1 called the mesh size, the pattern vectors are

constructed as Œ0 1, Œ1 0, Œ 1 0, and Œ0 1; they may be called direction vectors.

Then the PS algorithm adds the direction vectors to the initial point X0 to compute the

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X0C Œ1 0I X0C Œ0 1

X0C Œ 1 0I X0C Œ0 1

Figure 1 illustrates the formation of the mesh and pattern vectors. The algorithm computes the objective function at the mesh points in the order shown.

The algorithm polls the mesh points by computing their objective function values

until it finds one whose value is smaller than the objective function value of X0. If

there is such a point, then the poll is successful and the algorithm sets this point equal to X1.

After a successful poll, the algorithm steps to iteration 2 and multiplies the current mesh size by 2. (This is called the expansion factor and has a default value of 2.) The mesh at iteration 2 contains the following points: 2 Œ1 0 C X1, 2 Œ0 1 C X1,

2 Œ 1 0 C X1, and 2 Œ0 1 C X1. The algorithm polls the mesh points until it finds

one whose value is smaller the objective function value of X1. The first such point it

finds is called X2, and the poll is successful. Because the poll is successful, the algorithm

multiplies the current mesh size by 2 to get a mesh size of 4 at the third iteration because the expansion factor D 2.

Second: If in iteration 3 (mesh size D 4) none of the mesh points have a smaller objective function value than the value at X2, the poll is declared unsuccessful. In this case the

algorithm does not change the current point at the next iteration, which is X3D X2. At

the next iteration, the algorithm multiplies the current mesh size by 0.5, a contraction factor, so that the mesh size at the next iteration is smaller. The algorithm then polls with a smaller mesh size.

The PS optimization algorithm will repeat the illustrated steps until it finds the optimal solution for the minimization of the objective function. The algorithm stops when any of the following conditions occurs:

the mesh size is less than mesh tolerance,

the number of iterations performed by the algorithm reaches a predefined maxi-mum,

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the total number of objective function evaluations performed by the algorithm reaches a predefined maximum,

the distance between the point found at one successful poll and the point found at the next successful poll is less than a predefined tolerance, or

the change in the objective function from one successful poll to the next successful poll is less than a predefined function tolerance.

Constraint Handling

Many ideas were suggested to insure that the solution will satisfy the constraints [24]. Currently, the only minor weakness in the nonlinear constraints handling procedure is due to the lack of specific information related to first-order derivatives. Although many systematic approaches have been employed to handle the nonlinear constraints, the augmented Lagrangian approach, which has been utilized by PS method, has overcome such shortcomings in dealing with nonlinear constraints. Lewis and Torczon [25] stated that, despite the absence of an explicit estimation of any derivatives (a characteristic of PS methods), PS-augmented Lagrangian approach exhibits all of the first-order convergence properties of the original algorithm advocated by Conn, Gould, and Toint [26, 27]. Lewis and Torczon [25] were able to overcome such drawbacks by proceeding with successive, inexact minimization of the augmented Lagrangian via PS methods, even without knowing exactly how inexact the minimization is. As a result, the size of the problem will increase by introducing new parameters.

The augmented Lagrangian pattern search (ALPS) has been used to solve nonlinear constraint problems in PS algorithm. ALPS proceeds to solve a nonlinear optimization problem with nonlinear constraints, linear constraints, and bounds [25, 28–30]. The variables bounds and linear constraints are handled separately from nonlinear constraints, in which a sub-problem is constructed and solved (having the objective function and nonlinear constraint function) using the Lagrangian and the penalty factors. Such a sub-problem is minimized using a PS method, where the linear constraints and bounds are satisfied. ALPS starts with an initial value for the penalty parameter, where the PS algorithm minimizes a series of the subproblem, which estimates the original problem. If the required accuracy and feasibility conditions are met, then the Lagrangian estimates are updated. If not, a penalty factor is added to the penalty parameter. This, in turns, leads to a new formation of a subproblem and ultimately results in a new minimization problem. The above steps are repeated until one of the stopping criteria is reached. For further explanation on how PS handles constraints, refer to [25–27, 31].

4. Numerical Results

A set of Matlab files, incorporated in the GA and DS toolbox, implementing the proposed PS method, have been used to solve various ED problems. Thus, cost coefficients of the fuel cost and the combined objective function for all test cases have been coded in Matlab environment.

Initially, several runs have been carried out with different values of the key parameters of PS, such as the initial mesh size and the mesh expansion and contraction factors. In this study, the mesh size and the mesh expansion and contraction factors are selected as 1, 2, and 0.5, respectively. In addition, a vector of initial points, i.e., X0, was randomly

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guess for the PS to proceed. As for the stopping criteria, all tolerances were set to 10 6_,

and the maximum number of iterations and function evaluations set to 1000.

Case I: EDVP

This test case consists of 13 generating units with a quadratic cost function combined with the effects of valve-point loading. The unit’s data (upper and lower bounds) along with the cost coefficients for the fuel cost (a, b, c, e, and f ) for the 13 generators with valve-point loading are given in [32, 33]. Note that losses are ignored in this case.

The PS algorithm has been executed 50 times with different starting points to study its performance and effectiveness. The solution of the PS method and the execution times for the 50 runs were compared with the outcome of other evolutionary methods applied to the same test system as reported in [33]. This numerical experiment compares the performance of the PS with the other methods in terms of the dispatching costs and convergence speed. Table 1 shows the optimal solutions determined by PS for the 13 generators, while the execution time and cost comparison are shown in Table 2.

As can be seen from Table 2, the optimum solution of the PS is better than the solu-tions of all the other evolutionary methods. Moreover, the execution time is significantly shorter than for the other methods [33].

The convergence of the PS process is shown in Figure 2, where only 70 iterations were needed to locate the optimal solution. However, PS may be allowed to continue the search in the neighborhood of the optimal solution to increase the confidence in the result. In this case, the PS terminates after 52 further iterations (a total of 122).

Figure 3 illustrates the mesh size throughout the convergence process. It is apparent that the mesh size decreases until the algorithm terminates, in this case at a mesh size

Table 1

Generator loading and fuel cost determined by PS with total load demand of 1800 MW

Generator Generator production (MW) Pg1 538.5587 Pg2 224.6416 Pg3 149.8468 Pg4 109.8666 Pg5 109.8666 Pg6 109.8666 Pg7 109.8666 Pg8 109.8666 Pg9 109.8666 Pg10 77.4666 Pg11 40.2166 Pg12 55.0347 Pg13 55.0347 P Pgi D 1800 MW Total cost: $17,969.17

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Table 2

Comparison of PS, GA, and EP Evolution method Mean time (s) Best time (s) Mean cost ($) Maximum cost ($) Minimum cost ($) CEP 294.96 293.41 18,190.32 18,404.04 18,048.21 FEP 168.11 166.43 18,200.79 18,453.82 18,018.00 MFEP 317.12 315.98 18,192.00 18,416.89 18,028.09 IFEP 157.43 156.81 18,127.06 18,267.42 17,994.07 PS 5.88 1.65 18,088.84 18,233.52 17,969.17

CEP: Classical evolutionary programming; FEP: fast evolutionary programming; MFEP: modi-fied fast evolutionary programming; IFEP: improved fast evolutionary programming.

of 1.5259e–005, which is more than the stopping criteria, indicating that this particular run did not terminate on the mesh size tolerance. Figure 3 shows that the polling was initially successful, leading to enlarging the mesh size. At iteration number 8, the mesh size has decreased, indicating an unsuccessful poll in the previous iteration. This scenario continues throughout until reaching one of the termination criteria.

The PS has determined the optimal solution for 50 different initial points. The solutions of each run are shown in Figure 4. The total execution time for the 50 runs

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Figure 3. Convergence of PS mesh size for the 13 generating units.

was 294.06 s. For this test case, looking at the maximum, mean, and minimum costs in Table 2, it may be clearly seen that PS outperforms GA and evolutionary programming (EP) by reaching a lower value of the maximum and mean costs for 50 different runs.

Case II: MAED

The MAED problem considered consists of four areas with tie lines connecting these areas. Each area contains four generation units. Note that quadratic cost functions are used to model the cost of generation F .Pg i/, but the tie line transmission costs, G.tj k/,

are assumed to be linear functions of the power transfer. The generators’ data and the tie lines’ coefficients, along with their limits, are all given in [14].

Different heuristic methods, such as particle swarm optimization (PSO) and EP, have been applied to the same problem, and the results are reported in [34, 35]. The results using the PS and other methods are shown in Tables 3 and 4. The optimal solution obtained by PS is obviously better than those obtained by various EP algorithms and is similar to the result obtained by network flow programming (NFP) [14], which is not heuristic in nature. In addition, the computation time of PS is less than the execution times of the variants of EP.

As illustrated in Figure 5, PS has located the optimal solution after only 200 iterations, but it continues computating and refining the result. Figure 6 shows the mesh size expansion and contraction behavior during the PS search for the global minimum.

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Figure 4. Objective function value for 50 different starting points.

Table 3

Comparison of PS and other heuristic methods (16 generators) Optimization techniques

Generator IFEP FEP CEP NFP PS

Area 1 P1 (MW) 149.998 149.997 150.000 150.000 150.000 Demand P2 (MW) 099.986 099.968 100.000 100.000 100.000 400 MW P3 (MW) 068.270 067.017 068.826 066.970 066.971 P4 (MW) 099.940 099.774 099.985 100.000 100.000 Area 2 P5 (MW) 056.349 057.181 056.373 056.970 56.9718 Demand P6 (MW) 096.753 095.554 093.519 096.250 96.2518 200 MW P7 (MW) 041.264 041.736 042.546 041.870 41.8718 P8 (MW) 072.586 072.748 072.647 072.520 72.5218 Area 3 P9 (MW) 050.003 050.030 050.000 050.000 050.002 Demand P10 (MW) 035.985 036.552 036.399 036.270 036.272 350 MW P11 (MW) 038.012 038.413 038.323 038.490 038.492 P12 (MW) 037.426 037.001 036.903 037.320 037.322 Area 4 P13 (MW) 149.988 149.986 150.000 150.000 150.000 Demand P14 (MW) 099.964 099.995 100.000 100.000 100.000 300 MW P15 (MW) 057.601 057.568 056.648 057.050 057.051 P16 (MW) 095.874 096.482 095.826 096.270 096.271

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Table 4

Comparison of PS and other heuristic methods (tie lines)

Area Tie lines values (MW)

From To IFEP FEP CEP NFP PS

1 2 00.094 00.062 00.000 00.000 00.000 1 3 18.649 18.241 19.587 18.180 18.181 1 4 00.000 00.000 00.000 00.000 00.000 2 1 00.018 00.000 00.018 00.000 00.000 2 3 69.997 69.790 68.861 69.730 69.730 2 4 00.000 00.000 00.000 00.000 00.000 3 1 00.000 00.000 00.000 00.000 00.000 3 2 00.000 00.000 00.000 00.000 00.000 3 4 00.000 00.000 00.000 00.000 00.000 4 1 00.549 01.548 00.758 01.210 01.210 4 2 02.951 02.509 01.789 02.110 02.111 4 3 99.927 99.974 99.927 100.00 100.00 Total cost ($/h) 7337.51 7337.52 7337.75 7337.0 7336.98 Computation time (s) 23.97 7.47 7.82/11.49 — 5.77 Population size 100 100 100 — — No. of iterations 585 645 758/920 — 1225

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Figure 6. Mesh size.

Case III: CEED

In this combined environmental ED case, a six-generator system is considered.

Infor-mation about the generators’ fuel cost, NOx emission functions, the B matrix, loss

coefficients, and the operating limits are detailed in [36]. The total load demand is set to 700 MW, and the weighting factor is 0.5.

The PS results of the line losses, emission, fuel cost, total cost, and computation time are presented and compared with results of other heuristic methods (GA and EP from [34, 35]) in Table 5. It can be seen that PS has reached the best total and fuel costs, and also has produced the best time of computation compared with the other methods. In addition, PS came third in line losses and emissions. The convergence of the PS needs only 40 iterations and 2.05 s to reach the optimal solution, which are significantly less for EP and GA.

Case IV: QCFED

According to [17], it is an industry practice to adopt a cubic polynomial for modeling fuel costs of generation units. This is particularly important in situations with generation units having non-monotonically increasing incremental curves [19]. In this case, a three-generator system is considered with third-order cost functions. Information about the generators’ fuel-cost coefficients, the B matrix, loss coefficients, and the operating limits are detailed in [18].

The optimal solution of PS is given in Table 6, along with the results obtained by conventional DP from [18] for comparison purposes. Clearly, the PS has converged to a better solution, while the execution time is less than 1 s. Also, the significant reduction in line losses is obvious.

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Table 5

Comparison of PS and other heuristic methods

Optimization techniques

Generator IFEP FEP CEP FCGA PS P1 (MW) 077.142 077.358 077.274 080.16 77.4318 P2 (MW) 049.925 049.669 049.639 053.71 048.894 P3 (MW) 048.764 048.316 048.535 040.93 048.516 P4 (MW) 103.486 104.369 103.525 116.23 104.5679 P5 (MW) 259.805 260.663 260.695 251.20 260.8632 P6 (MW) 191.828 190.473 191.233 190.62 190.6723 Line losses (MW) 30.949 30.849 30.901 32.850 30.945 Emission (kg/h) 530.5164 532.5046 524.49 527.46 528.33 Fuel cost ($/h) 38,214.02 38,214.23 38,216.47 38,408.82 38,208.63 Total cost ($/h) 19,369.84 19,369.89 19,369.84 19,468.14 19,368.48 Computation time (s) 3.874 1.598 4.48 — 2.05 No. of iterations 57 65 77 — 40

FCGA: Fuzzy controlled genetic algorithm.

Table 6

Comparison of PS and DP with total demand 1400 MW Optimization technique Generator DP PS P1 (MW) 360.2 372.29 P2 (MW) 406.4 356.0 P3 (MW) 676.8 712.0 Line losses (MW) 43.4 40.29 Fuel cost ($/h) 6642.26 6639.01 Computation time (s) — 0.619 No. of iterations 5 6

5. Conclusions

This paper introduces a new approach based on PS optimization to solve various prob-lems of power system ELD. The proposed method has been applied to four types of problems, including the EDVP effects, MAED, CEED, and QCFED. Through extensive comparisons, it has been demonstrated that the PS approach outperforms other heuristic methods in terms of reaching a better optimal solution; at the same time, the simplicity of the PS process makes the algorithm computationally more efficient. However, the PS is more sensitive than the GA or EP to the initial guess (starting point) and appears to rely more heavily on how close the given initial point is to the expected optimum. This, in turn, makes the PS method more susceptible to getting trapped in local minima. Overall, the PS approach has been found to be a very efficient technique to study a wide range of optimization problem in the area of power system.

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Acknowledgment

This work has been supported by the government of The State of Kuwait, Public Authority of Applied Science & Training, Kuwait (Project # TS-06-02).

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